Rough numbers between consecutive primes

First things first: due to an abrupt suspension of NSF funding to my home university of UCLA, the Institute of Pure and Applied Mathematics (which had been preliminarily approved for a five-year NSF grant to run the institute) is currently fundraising to ensure continuity of operations during the suspension, with a goal of raising $500,000. Donations can be made at this page. As incoming Director of Special Projects at IPAM, I am grateful for the support (both moral and financial) that we have already received in the last few days, but we are still short of our fundraising goal.

Back to math. Ayla Gafni and I have just uploaded to the arXiv the paper “Rough numbers between consecutive primes“. In this paper we resolve a question of Erdös concerning rough numbers between consecutive gaps, and with the assistance of modern sieve theory calculations, we in fact obtain quite precise asymptotics for the problem. (As a side note, this research was supported by my personal NSF grant which is also currently suspended; I am grateful to recent donations to my own research fund which have helped me complete this research.)

Define a prime gap to be an interval between consecutive primes. We say that a prime gap contains a rough number if there is an integer whose least prime factor is at least the length of the gap. For instance, the prime gap contains the rough number , but the prime gap does not (all integers between and have a prime factor less than ). The first few for which the prime gap contains a rough number are

Numerically, the proportion of for which the prime gap does not contain a rough number decays slowly as increases:

Erdös initially thought that all but finitely many prime gaps should contain a rough number, but changed his mind, as per the following quote:

…I am now sure that this is not true and I “almost” have a counterexample. Pillai and Szekeres observed that for every , a set of consecutive integers always contains one which is relatively prime to the others. This is false for , the smallest counterexample being . Consider now the two arithmetic progressions and . There certainly will be infinitely many values of for which the progressions simultaneously represent primes; this follows at once from hypothesis H of Schinzel, but cannot at present be proved. These primes are consecutive and give the required counterexample. I expect that this situation is rather exceptional and that the integers for which there is no satisfying and have density .

In fact Erdös’s observation can be made simpler: any pair of cousin primes for (of which is the first example) will produce a prime gap that does not contain any rough numbers.

The latter question of Erdös is listed as problem #682 on Thomas Bloom’s Erdös problems website. In this paper we answer Erdös’s question, and in fact give a rather precise bound for the number of counterexamples:

Theorem 1 (Erdos #682) For , let be the number of prime gaps with that do not contain a rough number. Then

Assuming the Dickson–Hardy–Littlewood prime tuples conjecture, we can improve this to

for some (explicitly describable) constant .

In fact we believe that , although the formula we have to compute converges very slowly. This is (weakly) supported by numerical evidence:

While many questions about prime gaps remain open, the theory of rough numbers is much better understood, thanks to modern sieve theoretic tools such as the fundamental lemma of sieve theory. The main idea is to frame the problem in terms of counting the number of rough numbers in short intervals , where ranges in some dyadic interval and is a much smaller quantity, such as for some . Here, one has to tweak the definition of “rough” to mean “no prime factors less than ” for some intermediate (e.g., for some turns out to be a reasonable choice). These problems are very analogous to the extremely well studied problem of counting primes in short intervals, but one can make more progress without needing powerful conjectures such as the Hardy–Littlewood prime tuples conjecture. In particular, because of the fundamental lemma of sieve theory, one can compute the mean and variance (i.e., the first two moments) of such counts to high accuracy, using in particular some calculations on the mean values of singular series that go back at least to the work of Montgomery from 1970. This second moment analysis turns out to be enough (after optimizing all the parameters) to answer Erdös’s problem with a weaker bound

To do better, we need to work with higher moments. The fundamental lemma also works in this setting; one now needs precise asymptotics for the mean value of singular series of -tuples, but this was fortunately worked out (in more or less exactly the format we needed) by Montgomery and Soundararajan in 2004. Their focus was establishing a central limit theorem for the distribution of primes in short intervals (conditional on the prime tuples conjecture), but their analysis can be adapted to show (unconditionally) good concentration of measure results for rough numbers in short intervals. A direct application of their estimates improves the upper bound on to

and some more careful tweaking of parameters allows one to remove the error. This latter analysis reveals that in fact the dominant contribution to will come with prime gaps of bounded length, of which our understanding is still relatively poor (it was only in 2014 that Yitang Zhang famously showed that infinitely many such gaps exist). At this point we finally have to resort to (a Dickson-type form of) the prime tuples conjecture to get the asymptotic (2).